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%I #22 Sep 27 2023 02:43:10
%S 1,1,-2186,1,78126,-2186,-823542,1,4780783,78126,-19487170,-2186,
%T 62748518,-823542,-170783436,1,410338674,4780783,-893871738,78126,
%U 1800262812,-19487170,-3404825446,-2186,6103593751,62748518,-10455572420,-823542
%N a(n) = Sum_{d|n, d==1 mod 4} d^7 - Sum_{d|n, d==3 mod 4} d^7.
%H Seiichi Manyama, <a href="/A321823/b321823.txt">Table of n, a(n) for n = 1..10000</a>
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F a(n) = a(A000265(n)). - _M. F. Hasler_, Nov 26 2018
%F G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^7*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 06 2018
%F Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^7)^(e+1)-1)/(p^7-1) if p == 1 (mod 4) and ((-p^7)^(e+1)-1)/(-p^7-1) if p == 3 (mod 4). - _Amiram Eldar_, Sep 27 2023
%t s[n_, r_] := DivisorSum[n, #^7 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t f[p_, e_] := If[Mod[p, 4] == 1, ((p^7)^(e+1)-1)/(p^7-1), ((-p^7)^(e+1)-1)/(-p^7-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Sep 27 2023 *)
%o (PARI) apply( A321823(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^7), [1..40]) \\ _M. F. Hasler_, Nov 26 2018
%Y Column k=7 of A322143.
%Y Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%Y Cf. A000265.
%K sign,easy,mult
%O 1,3
%A _N. J. A. Sloane_, Nov 24 2018
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